When searching for synonyms for the phrase "quest after," there are many options to choose from depending on the context. Some possible alternatives include "pursue," "seek," "hunt," "chase," "explore," "probe," "investigate," "delve," "dig," "trace," "follow," "scan," "scrutinize," "track down," and "find out." These words all have slightly different connotations and nuances, so it is important to choose the right word for the situation. However, they all share the common meaning of actively looking for or seeking something, whether it be information, a person, or an object.